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How can I show that a given (not necessarily unit-speed) parametrization $\gamma(t)$ of a curve in $\mathbb{R}^3$ which exhibits zero curvature is a line ? What I know is that zero curvature means that $$ \kappa(t) = \frac{\|\langle\dot\gamma(t),\dot\gamma(t)\rangle \ddot\gamma(t) - \langle\dot \gamma(t),\ddot \gamma(t)\rangle\dot \gamma(t)\|}{\|\dot \gamma(t)\|^4} = 0 $$ from which I can deduce that $$ \langle\dot\gamma(t),\dot\gamma(t)\rangle \ddot\gamma(t) = \langle\dot \gamma(t),\ddot \gamma(t)\rangle\dot \gamma(t) $$ that is, $$ \ddot\gamma(t) = \frac{\langle\dot \gamma(t),\ddot \gamma(t)\rangle}{\|\dot\gamma(t)\|^2} \dot \gamma(t) \qquad \text{for all } t\,. $$ Somehow I am blind here - how does this tell me then that $\ddot \gamma(t)$ vanishes identically? For this is what I need to deduce that $\gamma(t)$ is a line ..

Many thanks for your hints!

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If you use arc length what is the relation between $\kappa$ and $\ddot\gamma$? – Sigur Jan 26 '13 at 18:45
Your last formula implies that acceleration is in the direction of velocity. (That formula is just the projection formula of a vector on another.) The second derivative does not valish, $\kappa$ does. – Maesumi Jan 26 '13 at 18:49
Here is a slightly different version of the comment by @Sigur: What is the derivative of $\dot\gamma/\lVert\gamma\rVert$? – Harald Hanche-Olsen Jan 26 '13 at 18:52
@Sigur The answer follows by definition if I use arclength. I was wondering how I can proceed in case I cannot assume this - there might not always be an explicit unit - speed reparametrization at my disposal. – harlekin Jan 26 '13 at 19:07
@HaraldHanche-Olsen do you mean $\dot \gamma / \| \dot \gamma\|$ ? – harlekin Jan 26 '13 at 21:08
up vote 2 down vote accepted

$\ddot \gamma(t)$ won't vanish in general, just think of $t\mapsto (t^2, 0,0)$. If you don't want to reparametrize $\gamma$, you will need to show that $\gamma(t) = f(t) v_0$ for some vector $v_0$ and a scalar function $f(t)$. To do this you could for instance argue that the unique solution to $$\dot v(t) = g(t) v(t), \; v(0) = v_0, \qquad \text{where } g(t) := \frac{\langle\dot \gamma(t), \ddot \gamma(t)\rangle}{\Vert \dot \gamma(t)\Vert^2}$$ is given by $v(t) = f(t)v_0$ for some function $f$.

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